This app is a clock widget, which displays the time as a binary number. It's not like the binary clocks you might have seen for sale which display the hours as one binary number and the minutes as another; this displays the minutes since 12:00 (noon or midnight) as a single binary number. The low bits are on the inner ring, the high bits on the outer ring, increasing counterclockwise, like this:
The widget is resizable on Android 3.1+, and can be used on the lockscreen on Android 4.2+. If you want to reconfigure it after initial setup, just touch it.
This widget used some source code from another widget, which is under the GPL, so this widget has to be under the GPL as well. So here's the source. Enjoy!
In addition, the app aggregates a couple of images which are under the Creative Commons BY license, meaning I have to give attribution, so here it is: both of them are from Flickr user cliff1066. The sapphire is cropped from a photo of the Logan Sapphire, the emerald from a photo of the Maximilian Emerald Ring. (The wood background is a photo of my own; you may use it under the same CC BY license.)
The app is available in Google Play, or you can download it right here.